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        <title>The Yoneda Lemma</title>

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                <h1 id="title" titleSize="">
                    The Yoneda Lemma
                </h1>
            
            <h1 id="the-yoneda-embedding-yo">The Yoneda Embedding $\yo$</h1>
<h2 id="statement--proof">Statement &amp; Proof</h2>
<p>Fix a locally-small category $C$. $\yo:C\to\catset$</p>
<div class="env" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/theorem.svg"><b class="envTitle">Theorem (Yoneda). </b>Let $C$ be a locally-small category. For every functor $F:C\to\catset$ and every object $c\in C$, we have a bijection
$$\begin{equation}
y\coloneqq y_{F,c}:\Hom(C(c,\slot),F)\to Fc\ \ \ \ \ \ \ \ \mathit{sending}\ \ \ \ \ \ \ \ \big(\eta:C(c,\slot)\to F\big)\mapsto\eta_c1_c,
\end{equation}$$
which is moreover natural in both $F$ and $c$.</div>

<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof (Bijection).</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent"><p>Define a function $t\coloneqq t_{F,c}:Fc\to\Hom(C(C,\slot),F)$ by sending an element $a\in Fc$ to the natural transformation $t(a):C(c,\slot)\to F$ via
$$\begin{equation}
t(a)_d:C(c,d)\to Fd\ \ \ \ \ \ \ \ \textrm{mapping}\ \ \ \ \ \ \ \ (f:c\rightarrow d)\mapsto Ff(a)\in Fd
\end{equation}$$
for each $d\in C$. That $a=y(t(a))$ is immediate since $t(a)_c(1_c)=F(1_c)a=1_ca=a$.</p>
<p><br>  Conversely, let $\eta:C(c,\slot)\to F$ be a natural transformation, whence the diagram
<img class="tikz noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/img/yoneda_bijection.svg" width="300">
commutes for any morphism $f:c\to d$. In particular, chasing $1_c\in C(c,c)$ around gives us $Ff(\eta_c(1_c))=\eta_d(f_\ast1_c)$, and thus $t(y(\eta))_df=Ff(y(\eta))=\eta_df$ for all $f:c\to d$. Hence, we have $t(y(\eta))=\eta$, as desired.<span style="float:right;">$\blacksquare$</span></p>
</span></div>

<div class="space"></div>
<div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof (Naturality).</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">The naturality claims amounts to</span></div>

<h2 id="extensions-and-corollaries">Extensions and Corollaries</h2>
<div class="env" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/theorem.svg"><b class="envTitle">Theorem (Yoneda Embedding). </b>For any locally-small category $C$, the Yoneda embedding $\mf{y}:C^\textrm{op}\to\catset^C$ is fully-faithful.</div>

<h1 id="interpretations">Interpretations</h1>
<h2 id="group-actions">Group Actions</h2>


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                May 29, 2024 | Zhaoshen Zhai

                <div>References:
                    
                        <div class="metaButton metaRef listenDark">#Mac10</div>
                    
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                <div>Tags:
                    
                        <div class="metaButton metaTag listenDark">#Category_Theory</div>
                    
                        <div class="metaButton metaTag listenDark">#In_Progress</div>
                    
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